Two small aluminum spheres, each having mass $0.0250$ kg, are separated by $80.0$ cm. How many electrons does each sphere contain? (The atomic mass of aluminum is $26.982$ g/mol, and its atomic number is $13$.)

Verified step by step guidance

1. First, determine the number of moles of aluminum in each sphere. Use the formula: \( \text{moles} = \frac{\text{mass of sphere}}{\text{molar mass of aluminum}} \). Convert the mass of the sphere from kg to g by multiplying by 1000.
2. Calculate the number of moles: \( \text{moles} = \frac{25.0 \text{ g}}{26.982 \text{ g/mol}} \).
3. Next, use Avogadro's number to find the total number of aluminum atoms in each sphere. Avogadro's number is \( 6.022 \times 10^{23} \text{ atoms/mol} \). Multiply the number of moles by Avogadro's number.
4. Since each aluminum atom contains 13 electrons (as given by the atomic number), multiply the total number of atoms by 13 to find the total number of electrons in each sphere.
5. Finally, express the result as the total number of electrons in each sphere, ensuring all units are consistent and calculations are clear.

Key Concepts

Atomic Structure

Atoms are the basic units of matter, consisting of a nucleus surrounded by electrons. The atomic number of an element, like aluminum, indicates the number of protons in its nucleus, which is equal to the number of electrons in a neutral atom. Understanding atomic structure is crucial for determining the number of electrons in a given mass of an element.

Mole Concept

The mole is a fundamental unit in chemistry that represents a specific number of particles, typically atoms or molecules. One mole contains Avogadro's number of particles, approximately 6.022 x 10^23. This concept allows us to convert between the mass of a substance and the number of atoms or molecules it contains, which is essential for calculating the number of electrons in the spheres.

Mass-to-Mole Conversion

To find the number of moles in a given mass of a substance, we use the formula: moles = mass (g) / molar mass (g/mol). For aluminum, with a molar mass of 26.982 g/mol, this conversion helps determine how many moles of aluminum are present in the spheres, which is necessary to calculate the total number of electrons.